Fourier Series Fourier Sine Series Fourier Cosine Series Fourier - - PowerPoint PPT Presentation

SLIDE 1

Fourier Series

• Fourier Sine Series
• Fourier Cosine Series
• Fourier Series

– Convergence of Fourier Series for 2T -Periodic Functions – Convergence of Half-Range Expansions: Cosine Series – Convergence of Half-Range Expansions: Sine Series

• Sawtooth Wave
• Triangular Wave
• Parseval’s Identity and Bessel’s Inequality
• Complex Fourier Series
• Dirichlet Kernel and Convergence

SLIDE 2

Fourier Sine Series

• Definition. Consider the orthogonal system {sin
• nπx

T

• }∞

n=1 on [−T, T ]. A Fourier

sine series with coefficients {bn}∞

n=1 is the expression

F (x) =

∞

• n=1

bn sin

nπx

T

• Theorem. A Fourier sine series F (x) is an odd 2T -periodic function.
• Theorem. The coefficients {bn}∞

n=1 in a Fourier sine series F (x) are determined by the

formulas (inner product on [−T, T ])

bn =

• F, sin
• nπx

T

• sin
• nπx

T

• , sin
• nπx

T

= 2

T

T

F (x) sin

nπx

T

• dx.

SLIDE 3

Fourier Cosine Series

• Definition. Consider the orthogonal system {cos
• mπx

T

• }∞

m=0 on [−T, T ]. A Fourier

cosine series with coefficients {am}∞

m=0 is the expression

F (x) =

∞

• m=0

am cos

mπx

T

• Theorem. A Fourier cosine series F (x) is an even 2T -periodic function.
• Theorem. The coefficients {am}∞

m=0 in a Fourier cosine series F (x) are determined by

the formulas (inner product on [−T, T ])

am =

• F, cos
• mπx

T

• cos
• mπx

T

• , cos
• mπx

T

=   

2 T

T

0 F (x) cos

• mπx

T

• dx m > 0,

1 T

T

0 F (x) dx

m = 0.

SLIDE 4

Fourier Series

• Definition. Consider the orthogonal system {cos
• mπx

T

• }∞

m=0, {sin

• nπx

T

• }∞

n=1 , on

[−T, T ]. A Fourier series with coefficients {am}∞

m=0, {bn}∞ n=1 is the expression

F (x) =

∞

• m=0

am cos

mπx

T

• +

∞

• n=1

bn sin

nπx

T

• Theorem. A Fourier series F (x) is a 2T -periodic function.
• Theorem. The coefficients {am}∞

m=0, {bn}∞ n=1 in a Fourier series F (x) are determined

by the formulas (inner product on [−T, T ])

am =

• F, cos
• mπx

T

• cos
• mπx

T

• , cos
• mπx

T

=   

1 T

T

−T F (x) cos

• mπx

T

• dx m > 0,

1 2T

T

−T F (x) dx

m = 0. bn =

• F, sin
• nπx

T

• sin
• nπx

T

• , sin
• nπx

T

= 1

T

T

−T

F (x) sin

nπx

T

• dx.

SLIDE 5

Convergence of Fourier Series for 2T -Periodic Functions The Fourier series of a 2T -periodic piecewise smooth function f(x) is

a0 +

∞

• n=1
• an cos

nπx

T

• + bn sin

nπx

T

• where

a0 = 1 2T

T

−T

f(x)dx, an = 1 T

T

−T

f(x) cos

nπx

T

• dx,

bn = 1 T

T

−T

f(x) sin

nπx

T

• dx.

The series converges to f(x) at points of continuity of f and to f(x+)+f(x−)

2

• therwise.

SLIDE 6

Convergence of Half-Range Expansions: Cosine Series The Fourier cosine series of a piecewise smooth function f(x) on [0, T ] is the even 2T - periodic function

a0 +

∞

• n=1

an cos

nπx

T

• where

a0 = 1 T

T

f(x)dx, an = 2 T

T

f(x) cos

nπx

T

• dx.

The series converges on 0 < x < T to f(x) at points of continuity of f and to

f(x+)+f(x−) 2

• therwise.

SLIDE 7

Convergence of Half-Range Expansions: Sine Series The Fourier sine series of a piecewise smooth function f(x) on [0, T ] is the odd 2T - periodic function

∞

• n=1

bn sin

nπx

T

• where

bn = 2 T

T

f(x) sin

nπx

T

• dx.

The series converges on 0 < x < T to f(x) at points of continuity of f and to

f(x+)+f(x−) 2

• therwise.

SLIDE 8

Sawtooth Wave

• Definition. The sawtooth wave is the odd 2π-periodic function defined on −π ≤ x ≤

π by the formula sawtooth(x) =

        

1 2(π − x)

0 < x ≤ π,

1 2(−π − x)

−π ≤ x < 0, x = 0.

• Theorem. The sawtooth wave has Fourier sine series

sawtooth(x) =

∞

• n=1

1 n sin nx.

SLIDE 9

Triangular Wave

• Definition. The triangular wave is the even 2π-periodic function defined on −π ≤

x ≤ π by the formula twave(x) =

π − x

0 < x ≤ π, π + x −π ≤ x ≤ 0.

• Theorem. The triangular wave has Fourier cosine series

twave(x) = π 2 + 4 π

∞

• k=0

1 (2k + 1)2 cos(2k + 1)x.

SLIDE 10

ParsevaL’s Identity and Bessel’s Inequality

• Theorem. (Bessel’s Inequality)

a2

0 + 1

2

∞

• n=1

a2

n + b2 n

≤ 1

2T

T

−T

|f(x)|2dx

• Theorem. (Parseval’s Identity)

1 2T

T

−T

|f(x)|2dx = a2

0 + 1

2

∞

• n=1

a2

n + b2 n

• Theorem. Parseval’s identity for the sawtooth function implies

π2 12 = 1 2

∞

• n=1

1 n2

SLIDE 11

Complex Fourier Series

• Definition. Let f(x) be 2T -periodic and piecewise smooth. The complex Fourier series
• f f is

∞

• n=−∞

cne

inπx T ,

cn = 1 2T

T

−T

f(x)e

−inπx T

dx

• Theorem. The complex series converges to f(x) at points of continuity of f and to

f(x+)+f(x−) 2

• therwise.
• Theorem. (Complex Parseval Identity)

1 2T

T

−T

|f(x)|2dx =

∞

• n=−∞

|cn|2

SLIDE 12

Dirichlet Kernel and Convergence

• Theorem. (Dirichlet Kernel Identity)

1 2 + cos u + cos 2u + · · · + cos nu = sin

• n + 1

2

• u
• 2 sin
• 1

2u

• Theorem. (Riemann-Lebesgue)

For piecewise continuous g(x), lim

N→∞

π

−π

g(x) sin(Nx)dx = 0.

Proof: Integration theory implies it suffices to establish the result for smooth g. Integrate by parts to

• btain 1

n(g(−π) − g(π))(−1)n + 1 n

π

−π g(x) cos(nx)dx. Letting n → ∞ implies the result.

• Theorem. Let f(x) be 2π-periodic and smooth on the whole real line. Then the Fourier

series of f(x) converges uniformly to f(x).

SLIDE 13

Convergence Proof STEP 1. Let sN(x) denote the Fourier series partial sum. Using Dirichlet’s kernel for- mula, we verify the identity

f(x) − sN(x) = 1 π

x+π

x−π

(f(x) − f(x + w))

• sin((N + 1/2)w)

2 sin(w/2)

• dw

STEP 2. The integrand I is re-written as

I = f(x) − f(x + w) w w 2 sin(w/2) sin((N + 1/2)w).

STEP 3. The function g(w) = f(x) − f(x + w)

w w sin(w/2)

is piecewise continu-

• us. Apply the Riemann-Lebesgue Theorem to complete the proof of the theorem.

SLIDE 14

Gibbs’ Phenomena Engineering Interpretation: The graph of f(x) and the graph of

a0 + N

n=1(an cos nx + bn sin nx)

are identical to pixel resolution, provided N is sufficiently large. Computers can therefore graph f(x) using a truncated Fourier series. If f(x) is only piecewise smooth, then pointwise convergence is still true, at points of continuity of f, but uniformity of the convergence fails near discontinuities of f and f ′. Gibbs discovered the fixed-jump artifact, which appears at discontinuities of f.